Spread Spectrum Signals for Digital Communication: Direct Sequence Spread Binary PSK

Spread Spectrum Signals for Digital Communication
Manjunatha. P
manjup.jnnce@gmail.com
Professor
Dept. of ECE
J.N.N. College of Engineering, Shimoga
December 15, 2015

Overview Overview
Spread Spectrum Signals for Digital Communication
1 Model of Spread Spectrum Digital Communication System
2 Direct Sequence Spread Spectrum Signals
3 Frequency-Hopped Spread Spectrum Signals
4 CDMA
5 Time-hopping SS
6 Synchronization of SS systems
Manjunatha. P (JNNCE) Spread Spectrum Signals for Digital Communication December 15, 2015 2 / 70

Generation of Pseudo Noise codes Generation of Pseudo Noise codes
Generation of Pseudo Noise codes

Spread Spectrum Modulation (SSM) Background
Used in military for the past 50 years
Its commercial use started in 1980
Frequency Hopping: Invented in 1940 by Hedy Lamarr
Figure: Hedy Lamarr and George Antheil [1]

Figure: Hedy Lamarr invention details [2]
Spread spectrum modulation was originally developed for military to be used in the battle ground and in
the hostile territories where the enemy always tries to intrude into the communication system of the
friendly forces to steal information and to jam the systems.

Definition: Spread spectrum is a means of transmission in which the signal
occupies a bandwidth in excess of the minimum necessary to send the information:
the band spread is accomplished by means of a code which is independent of the
data, and synchronized reception with the code at the receiver is used for
de-spreading and subsequent data recovery.
Applications and advantages of spread spectrum systems
Low density power spectra for signal (To avoid being detected.)
To prevent eavesdropping.
To prevent the jamming of signals.
Improved interference rejection
CDMA applications
Secure communication
High resolution ranging
Antijam capability
Increased capacity and spectral efficiency in some mobile-cellular pesonal
communication system applications
Lower cost of implementation
Readily available IC components

A type of modulation in which the modulated signal bandwidth is much
greater than the message signal bandwidth.
The spreading of the message signal spectrum is done by a spreading code
called Pseudo Noise Code (PN Code) which is independent of the message
signal.

Generation of Pseudo Noise codes Types of Spread Spectrum
Types of Spread Spectrum
Direct Sequence Spread Spectrum (DS/SS)
Frequency Hopping Spread Spectrum (FH/SS)
1 Slow Frequency Hopping Spread Spectrum
2 Fast Frequency Hopping Spread Spectrum
Time Hopping Spread Spectrum (TH/SS)
Hybrid spread spectrum methods

Spreading Codes Spreading Codes
Spreading Codes
Maximal length sequences
Good auto and cross correlation, small code set
Gold codes and Kasami sequences
Are derived from M-sequences with similar correlation properties, and a larger
code set.
Walsh and Hadamard sequences
zero correlation between codes when aligned cross-correlation non-zero when
time shifted fixed spreading factor (codes of different length are not
orthogonal)

Pseudo Noise (PN) Sequence
Flip-Flop Flip-Flop
Flip-Flop
x1
x1
x3
x2
Output
Sequence
Modulo 2 adder
100, 110, 111, 011 101, 010, 001, 100,
0011101
N = 2m
− 1

A pseudo noise (PN) sequence is defined as a coded sequence of 1s and 0s with a certain
autocorrelation properties.
The class of sequence used in spread spectrum communication is usually periodic.
The major tasks of PN sequences are:
1 Spreading the bandwidth of the modulated signal to the larger bandwidth.
2 Distinguishing between the different user signals utilizing the same transmission bandwidth.
The maximum length sequence is a type of cyclic code represents a commonly used PN sequence.
Flip-Flop Flip-Flop
Flip-Flop
x1
x1
x3
x2
Output
Sequence
Modulo 2 adder
Consider the initial state of the shift register is 100 (i.e, x1=1, x2=0 and x3=0).
Then, the succession of state will be as follows: 100, 110, 111, 011,101, 010, 001, 100
The output of the sequence (the last position of each of the shift register) is therefore 0011101. The
choice of initial state 100 is an arbitrary one.
Any of the other six states could serve equally well as an initial state.

1 Property 1: Balance Property
In each period of a maximum-length sequence, the number of 1s is always one more than the number of
0s.
2 Property2: Run Property
Among the runs of 1s and of 0s in each period of maximum-length sequence, one-half the runs of each
kind are of length one, one-fourth are of length two, one-eighth are of length three, and so on as long as
these fractions represent meaningful numbers of runs
3 Property3: Autocorrelation property
The autocorrelation function of a maximum-length sequence is periodic and binary valued.
Property3: Autocorrelation property
It is a measure of similarity between a signal f (t) and τ second time-shifted replica of itself.
The autocorrelation function is plot of autocorrelation over all shifts (−t) of the signal.
Autocorrelation Ra(τ) in general is defined as:
Ra(τ) =
∞
Z
−∞
f (t)f (t − τ)dt
The autocorrelation function of a maximum length sequence is periodic and binary valued. This property is
called as correlation property.
Rc (τ) =
1
L
L−1
X
i=0
Ci ∗ C(i+τ) mod L τ = 0, 1, . . . L − 1
If symbol 1 and 0 are represented by +1 volt and -1 volt then autocorrelation has only two values.
Rc (τ) =

1 for τ = kL k = 0, 1, 2, ...
− 1
L for τ 6= kL k = 0, 1, 2, ...

Flip-
Flop
Flip-
Flop
Flip-
Flop
x4
x3
x2
Output
Sequence
Modulo-2
adder
Flip-
Flop
x1
1000, 0100, 0010, 1001, 1100, 0110, 1011, 0101, 1010, 1101, 1110, 1111,
0111, 0011, 0001, 1000,
000100110101111
N = 2m
− 1
000100110101111

0011101 =⇒ C0C1C2C3C4C5C6 C0C1C5=-1 C2C3C4C6=1
Rc (0) =
1
7
C2
0 + C2
1 + C2
2 + C2
3 + C2
4 + C2
5 + C2
6

= 1
Rc (1) =
1
7
(C0.C6 + C1.C0 + C2.C1 + C3.C2 + C4.C3 + C5.C4 + C6.C5) = −
1
7
=
1
7
(−1.1 + −1. − 1 + 1. − 1 + 1.1 + 1.1 + −1.1 + 1. − 1) = −
1
7
Rc
(t)
0 0 1 1 1 0 1 0 0 1 1 1 0 1
Tc NTc
Tc
Tc

Spreading Codes Gold codes
Gold codes
Combining two m-sequences creates Gold codes.
Gold sequences are an important class of sequences that allow construction of
long sequences with three valued Auto Correlation Function ACFs.
Gold sequences are constructed from pairs of preferred m-sequences by
modulo-2 addition of two maximal sequences of the same length.
Gold sequences are in useful in non-orthogonal (asynchronous) CDMA.
The use of Gold sequences permits the transmission to be asynchronous.
The receiver can synchronize using the auto-correlation property of the Gold
sequence.

Spreading Codes Walsh Codes
Walsh Codes
There is a simple method on how to generate a form of orthogonal Walsh
codes.
H0 =

1 1
1 −1

Hk =

Hk−1 Hk−1
Hk−1 −Hk−1


Notion of spread spectrum Notion of spread spectrum
Notion of spread spectrum
m(t)
c(t)
b(t)
r(t)
i(t)
m(t)
z(t)
c(t)
r(t)
Baseband
LPF
m(t) = c(t)b(t)
m(t)
c(t)
b(t)
r(t)
i(t)
m(t)
z(t)
c(t)
r(t)
Baseband
LPF
r(t) = m(t) + i(t) = c(t)b(t) + i(t)
m(t)
c(t)
b(t)
r(t)
i(t)
m(t)
z(t)
c(t)
r(t)
Baseband
LPF z(t) = c(t)r(t) = c2
(t)b(t)+c(t)i(t)
c2
(t) = 1
z(t) = b(t) + c(t)i(t)

Notion of spread spectrum Notion of spread spectrum
Data b(t)
Spreading Code c(t)
Product Signal m(t)
Tb
NTc
0 0 1 1 1 0 1

Direct Sequence Spread Binary PSK Direct Sequence Spread Binary PSK
Direct Sequence Spread Binary PSK
Phase
modulator
0
2 cos( ( ))
P w t t
θ
+
( )
c t
0
2 cos( )
P w t
Binary
Data
0
2 cos( ( )
( )
)
P w t
c t t
θ
+
Band pass
filter
0
2 cos( )
P w t
Despreading
Mixer
( )
d
c t T
−
0
( )
2 cos( (
) )
( )
d d
P w t i
t
c t T T t
θ
− + +
−
Phase
demodulator
Estimated
data
Figure: DSS Transmitter
Phase
modulator
0
2 cos( ( ))
P w t t
θ
+
( )
c t
0
2 cos( )
P w t
Binary
Data
0
2 cos( ( )
( )
)
P w t
c t t
θ
+
Band pass
filter
0
2 cos( )
P w t
Despreading
Mixer
( )
d
c t T
−
0
( )
2 cos( (
) )
( )
d d
P w t i
t
c t T T t
θ
− + +
−
Phase
demodulator
Estimated
data
Figure: DSS Receiver

Figure: (a) m(t)=c(t).b(t) (b)Carrier (c) DS-BPSK signal

Figure: Bandwidth

Model of Spread Spectrum Digital Communication Model of Spread Spectrum Digital Communication
Model of Spread Spectrum Digital Communication

Model of Spread Spectrum Digital Communication Model of Spread Spectrum Digital Communication
The channel encoder and decoder and the modulator and demodulator are the basic elements of the
system.
There are two identical pseudorandom pattern generator one with modulator at the transmitting end
and a second with the demodulator at the receiving end.
The synchronization of the PN sequence generated at the receiver with the PN sequence contained in
the incoming received signal is required in order to demodulate the received signal
Channel
Encoder
Channel
Channel
Decoder
PN
sequence
generator
Modulator
PN
sequence
generator
Information
sequence
Output
Demodulator
k bits
m bits
Figure: Model of spread spectrum communication system

Direct sequence spread spectrum signals Direct sequence spread spectrum signals
Direct sequence spread spectrum signals

PN
Generator
Mod-2
adder
Balanced
Modulator
PN
Generator
Mod-2
adder
Balanced
Modulator
Encoder
Local
oscillator
Adder
QPSK
signal
( )
cos 2 c
f t
π
( )
sin 2 c
f t
π
Data
Figure: Direct sequence spread spectrum QPSK modulator
The message bits are modulo-2 added by PN sequence and if
bi is the ith bit of the PN sequence and ci is the corresponding
bit from the encoder, the modulo-2 sum is
ai = bi ⊕ ci
From this, ai = 0 when bi = ci and ai = 1 when bi 6= ci .
The sequence ai is mapped into a binary PSK signal of the
form s(t) = ±Re[g(t)ej2πfc t
] as follows:
gi (t) =

g(t − iTc ) (ai = 0)
−g(t − iTc ) (ai = 1)
s(t) = ±Re[g(t)e
j2πfc t
] [e
±jx
= cosx ± jsinx] Figure: Illustrates the spreading process

The modulo-2 addition can also be implemented by multiply-
ing two waveforms as follows. The coded sequence is multip-
ied by
ci (t) = (2ci − 1)g(t − iTc )
pi (t) = (2bi − 1)p(t − iTc )
where p(t) is the rectangular pulse of duration Tc . The
equivalent low pass transmitted signal for the ith
code is
gi (t) = pi (t)ci (t)
= (2bi − 1)(2ci − 1)g(t − iTc )
Figure: Illustrates the spreading process
1 0
1
1
1
1
1
1
1 0
0
0
0
0
0 0
Message
Modulated
Message
Spread
Message
Figure: Illustrates the spreading process

The received equivalent low pass signal for the ith
code is
ri (t) = pi (t)ci (t) + z(t)
= (2bi − 1)(2ci − 1)g(t − iTc ) + z(t)
where z(t) represents the interference or jamming
signal.
Matched
filter
Sample
Chip-rate
clock
PN code
generator
2 1
i
b −
( )
r t To dcoder
Chip-rate
clock
( )
r t To dcoder
PN code
generator
Sample
0
()
c
T
dt
∫
*( )
g t
( )
i
p t
Sample
Chip-rate
clock
PN code
generator
2 1
i
b −
( )
r t To dcoder
0
()
c
T
dt
∫
*( )
g t
Figure: DSSS Demodulator

Consider an information rate to the encoder is R bits/s and the available
channel bandwidth is W Hz.
The reciprocal of the R denoted by Tb, i.e., Tb = 1/R defines the duration of
a pulse corresponding to the transmission time of an information bit.
The PN code is at a rate of W times/s. The reciprocal of the W denoted by
Tc , defines the duration of a pulse, which is called chip, Tc is called chip
interval.
The bandwidth expansion factor W /R may be expressed as
Be =
W
R
=
Tb
Tc
(1)
In practical system the ratio Tb/Tc is an integer,
Le =
Tb
Tc
(2)
which is the number of chips per information bit.

Code Division Multiple Access (CDMA) Code Division Multiple Access (CDMA)
Code Division Multiple Access (CDMA)

CDMA

CDMA
Forward
Link
Reverse
Link
Mobile Station Base Station

Code Division Multiple Access (CDMA) CDMA
Initially adopted in North America, Proposed by Qualcomm, standardized as IS-95 by
Telecommunication Industry Association (TIA) for use in 800 MHz and 1900 MHz freq bands
Base station to Mobile (Forward (Down)link ) and mobile to BS (Reverse (UP) link) channel bandwidth
is of 1.25 M Hz.
Both forward and reverse links are DS spread having a chip rate of 1.2288X106
chips per second.
Forward Link
CELP coder generates a variable data rate of 9600, 4800, 2400, and 1200 bits/s, (based speech activity)
in the frame of 20 ms.
Data is encoded by a rate of 1/2, constraint length K=9 convolutional encoder.
For lower speech activity 4800, 2400, or 1200 bits/s the output symbols from the convolutional encoder
are repeated either twice, four times, or eight times so as to maintain a constant bit rate of 9600 bits/s.
Block interleaver is used to overcome the effects of burst errors that may occur during the transmission
through the channel.
Scrambler is used for Data Encryption to make call more secure.
Scrambler will randomizes data and prevents the transition of a long series of 1’s or 0’s
Block interleaver data rate of 19.2 kbps are scrambled by multiplication with long code with chip rate of
1.2288 M chips/s and is decimated by factor of 64 to 19.2 kchips/s.
The long code is used to identify a call of a MS on the forward and reverse links.
Hadmard or Walsh code sequence of length 64 is assigned to each channel.
64 orthogonal sequences are assigned to each BS (64 Channels), One channel is used to transmit pilot
signal, which is used measure the channel characteristics (includes signal strength and the carrier phase
offset).
Another channel is used to provide time synchronization. one channel for paging activity.
Remaining 61 channels for user.

Figure: Block diagram of IS-95 forward link
Each user data is multiplied by Hadmard sequence, the resulting sequence is spread by two PN
sequences one in-phase and other in quadrature phase of length N = 215
.
Different BS are identified by different offsets of these PN sequences.

D0 D5
D4
D3
D2
D1 D8
D7
D6
Possible
rates
9600 bps
4800 bps
2400 bps
1200 bps
9 Element Shift Register
XOR
XOR
752 octal
=111101011
561 octal
=101110001
G0 G1
Figure: Convolution Encoder rate 1/2 (r=input bits/output bits)

Code Division Multiple Access (CDMA) Reverse Link
Reverse Link
Figure: Block diagram of IS-95 reverse link
Orthogonal Modulation
6 bit block of data is mapped into 64 Hadamard seq = (64/6) ∗ 28.8 = 307.8
64-ary orthogonal modulation using the same Walsh function in the forward link
Contrary to the forward link, used for orthogonal data modulation
One Walsh function is transmitted for six coded bits
Modulated symbol rate: 28.8kbps ∗ 64chips/6codedbits = 307.2kcps
Increase interference tolerance

Code Division Multiple Access (CDMA) Processing Gain and Jamming Margin
Processing Gain and Jamming Margin: Some Basic Relations:
Information rate (Data rate) is R bits/s ∴ Bit duration Tb = 1/R
PN code generator data rate is W times/s ∴ Chip duration Tc = 1/W
Signal energy per bit Eb = Pav Tb = Pav /R
The average jamming power (Jav )is Jav = J0W where J0 PSD of jamming signal ∴ J0 = Jav /W
Eb
J0
=
Pav /R
Jav /W
=
W /R
Jav /Pav
=

PS /R
PI /W
=
W /R
PI /PS

Pav /Jav is the jamming to signal power ratio which is greater than unity, Jav /Pav is jamming margin.
The ratio W /R = Tb/Tc = Be = Lc is the bandwidth expansion factor or number of chips per info bit
is called processing gain.
Rc dmin is the coding gain and all these are related by:
(SNR)dB = (W /R)dB + (Rc dmin)dB − (Jav /Pav )dB
Probability of error for binary signaling system is
P2 = Q


s
2Eb
J0

 = Q
s
2W /R
Jav /Pav
!
Pe =
1
2
erfc


s
Eb
N0

 and erfc(u) = 1 − erf (u)
If there are Nu simultaneous users then the desired signal-to-nose interference ratio at a given receiver is
(Pav = PS Jav = PN )
PS
PN
=
PS
(Nu − 1)PS
=
1
Nu − 1

Complementary Error Function Table
x erfc(x) x erfc(x) x erfc(x) x erfc(x) x erfc(x) x erfc(x) x erfc(x)
0 1.000000 0.5 0.479500 1 0.157299 1.5 0.033895 2 0.004678 2.5 0.000407 3 0.00002209
0.01 0.988717 0.51 0.470756 1.01 0.153190 1.51 0.032723 2.01 0.004475 2.51 0.000386 3.01 0.00002074
0.02 0.977435 0.52 0.462101 1.02 0.149162 1.52 0.031587 2.02 0.004281 2.52 0.000365 3.02 0.00001947
0.03 0.966159 0.53 0.453536 1.03 0.145216 1.53 0.030484 2.03 0.004094 2.53 0.000346 3.03 0.00001827
0.04 0.954889 0.54 0.445061 1.04 0.141350 1.54 0.029414 2.04 0.003914 2.54 0.000328 3.04 0.00001714
0.05 0.943628 0.55 0.436677 1.05 0.137564 1.55 0.028377 2.05 0.003742 2.55 0.000311 3.05 0.00001608
0.06 0.932378 0.56 0.428384 1.06 0.133856 1.56 0.027372 2.06 0.003577 2.56 0.000294 3.06 0.00001508
0.07 0.921142 0.57 0.420184 1.07 0.130227 1.57 0.026397 2.07 0.003418 2.57 0.000278 3.07 0.00001414
0.08 0.909922 0.58 0.412077 1.08 0.126674 1.58 0.025453 2.08 0.003266 2.58 0.000264 3.08 0.00001326
0.09 0.898719 0.59 0.404064 1.09 0.123197 1.59 0.024538 2.09 0.003120 2.59 0.000249 3.09 0.00001243
0.1 0.887537 0.6 0.396144 1.1 0.119795 1.6 0.023652 2.1 0.002979 2.6 0.000236 3.1 0.00001165
0.11 0.876377 0.61 0.388319 1.11 0.116467 1.61 0.022793 2.11 0.002845 2.61 0.000223 3.11 0.00001092
0.12 0.865242 0.62 0.380589 1.12 0.113212 1.62 0.021962 2.12 0.002716 2.62 0.000211 3.12 0.00001023
0.13 0.854133 0.63 0.372954 1.13 0.110029 1.63 0.021157 2.13 0.002593 2.63 0.000200 3.13 0.00000958
0.14 0.843053 0.64 0.365414 1.14 0.106918 1.64 0.020378 2.14 0.002475 2.64 0.000189 3.14 0.00000897
0.15 0.832004 0.65 0.357971 1.15 0.103876 1.65 0.019624 2.15 0.002361 2.65 0.000178 3.15 0.00000840
0.16 0.820988 0.66 0.350623 1.16 0.100904 1.66 0.018895 2.16 0.002253 2.66 0.000169 3.16 0.00000786
0.17 0.810008 0.67 0.343372 1.17 0.098000 1.67 0.018190 2.17 0.002149 2.67 0.000159 3.17 0.00000736
0.18 0.799064 0.68 0.336218 1.18 0.095163 1.68 0.017507 2.18 0.002049 2.68 0.000151 3.18 0.00000689
0.19 0.788160 0.69 0.329160 1.19 0.092392 1.69 0.016847 2.19 0.001954 2.69 0.000142 3.19 0.00000644
0.2 0.777297 0.7 0.322199 1.2 0.089686 1.7 0.016210 2.2 0.001863 2.7 0.000134 3.2 0.00000603
0.21 0.766478 0.71 0.315335 1.21 0.087045 1.71 0.015593 2.21 0.001776 2.71 0.000127 3.21 0.00000564
0.22 0.755704 0.72 0.308567 1.22 0.084466 1.72 0.014997 2.22 0.001692 2.72 0.000120 3.22 0.00000527
0.23 0.744977 0.73 0.301896 1.23 0.081950 1.73 0.014422 2.23 0.001612 2.73 0.000113 3.23 0.00000493
0.24 0.734300 0.74 0.295322 1.24 0.079495 1.74 0.013865 2.24 0.001536 2.74 0.000107 3.24 0.00000460
0.25 0.723674 0.75 0.288845 1.25 0.077100 1.75 0.013328 2.25 0.001463 2.75 0.000101 3.25 0.00000430
0.26 0.713100 0.76 0.282463 1.26 0.074764 1.76 0.012810 2.26 0.001393 2.76 0.000095 3.26 0.00000402
0.27 0.702582 0.77 0.276179 1.27 0.072486 1.77 0.012309 2.27 0.001326 2.77 0.000090 3.27 0.00000376
0.28 0.692120 0.78 0.269990 1.28 0.070266 1.78 0.011826 2.28 0.001262 2.78 0.000084 3.28 0.00000351
0.29 0.681717 0.79 0.263897 1.29 0.068101 1.79 0.011359 2.29 0.001201 2.79 0.000080 3.29 0.00000328
0.3 0.671373 0.8 0.257899 1.3 0.065992 1.8 0.010909 2.3 0.001143 2.8 0.000075 3.3 0.00000306
0.31 0.661092 0.81 0.251997 1.31 0.063937 1.81 0.010475 2.31 0.001088 2.81 0.000071 3.31 0.00000285
0.32 0.650874 0.82 0.246189 1.32 0.061935 1.82 0.010057 2.32 0.001034 2.82 0.000067 3.32 0.00000266
0.33 0.640721 0.83 0.240476 1.33 0.059985 1.83 0.009653 2.33 0.000984 2.83 0.000063 3.33 0.00000249
0.34 0.630635 0.84 0.234857 1.34 0.058086 1.84 0.009264 2.34 0.000935 2.84 0.000059 3.34 0.00000232
0.35 0.620618 0.85 0.229332 1.35 0.056238 1.85 0.008889 2.35 0.000889 2.85 0.000056 3.35 0.00000216
0.36 0.610670 0.86 0.223900 1.36 0.054439 1.86 0.008528 2.36 0.000845 2.86 0.000052 3.36 0.00000202
0.37 0.600794 0.87 0.218560 1.37 0.052688 1.87 0.008179 2.37 0.000803 2.87 0.000049 3.37 0.00000188
0.38 0.590991 0.88 0.213313 1.38 0.050984 1.88 0.007844 2.38 0.000763 2.88 0.000046 3.38 0.00000175
0.39 0.581261 0.89 0.208157 1.39 0.049327 1.89 0.007521 2.39 0.000725 2.89 0.000044 3.39 0.00000163
0.4 0.571608 0.9 0.203092 1.4 0.047715 1.9 0.007210 2.4 0.000689 2.9 0.000041 3.4 0.00000152
0.41 0.562031 0.91 0.198117 1.41 0.046148 1.91 0.006910 2.41 0.000654 2.91 0.000039 3.41 0.00000142
0.42 0.552532 0.92 0.193232 1.42 0.044624 1.92 0.006622 2.42 0.000621 2.92 0.000036 3.42 0.00000132
0.43 0.543113 0.93 0.188437 1.43 0.043143 1.93 0.006344 2.43 0.000589 2.93 0.000034 3.43 0.00000123
0.44 0.533775 0.94 0.183729 1.44 0.041703 1.94 0.006077 2.44 0.000559 2.94 0.000032 3.44 0.00000115
0.45 0.524518 0.95 0.179109 1.45 0.040305 1.95 0.005821 2.45 0.000531 2.95 0.000030 3.45 0.00000107
0.46 0.515345 0.96 0.174576 1.46 0.038946 1.96 0.005574 2.46 0.000503 2.96 0.000028 3.46 0.00000099
0.47 0.506255 0.97 0.170130 1.47 0.037627 1.97 0.005336 2.47 0.000477 2.97 0.000027 3.47 0.00000092
0.48 0.497250 0.98 0.165769 1.48 0.036346 1.98 0.005108 2.48 0.000453 2.98 0.000025 3.48 0.00000086
0.49 0.488332 0.99 0.161492 1.49 0.035102 1.99 0.004889 2.49 0.000429 2.99 0.000024 3.49 0.00000080
Figure: Block diagram of IS-95 reverse link

Table I. Values of Erf(x) and Erfc(x)
(for positive values of x)
Value x Erf (x) Erfc(x)
0 0 1
0.05 0.0563720 0.9436280
0.1 0.1124629 0.8875371
0.15 0.1679960 0.8320040
0.2 0.2227026 0.7772974
0.25 0.2763264 0.7236736
0.3 0.3286268 0.6713732
0.35 0.3793821 0.6206179
0.4 0.4283924 0.5716076
0.45 0.4754817 0.5245183
0.5 0.5204999 0.4795001
0.55 0.5633234 0.4366766
0.6 0.6038561 0.3961439
0.65 0.6420293 0.3579707
0.7 0.6778012 0.3221988
0.75 0.7111556 0.2888444
0.8 0.7421010 0.2578990
0.85 0.7706681 0.2293319
0.9 0.7969082 0.2030918
0.95 0.8208908 0.1791092
1 0.8427008 0.1572992
1.1 0.8802051 0.1197949
1.2 0.9103140 0.0896860
1.3 0.9340079 0.0659921
1.4 0.9522851 0.0477149
1.5 0.9661051 0.0338949
1.6 0.9763484 0.0236516
1.7 0.9837905 0.0162095
1.8 0.9890905 0.0109095
1.9 0.9927904 0.0072096
2 0.9953223 0.0046777
2.1 0.9970205 0.0029795
2.2 0.9981372 0.0018628
2.3 0.9988568 0.0011432
2.4 0.9993115 0.0006885
2.5 0.9995930 0.0004070
2.6 0.9997640 0.0002360
2.7 0.9998657 0.0001343
2.8 0.9999250 0.0000750
2.9 0.9999589 0.0000411
3 0.9999779 0.0000221
3.1 0.9999884 0.0000116
3.2 0.9999940 0.0000060
3.3 0.9999969 0.0000031
3.4 0.9999985 0.0000015
3.5 0.9999993 0.0000007

Code Division Multiple Access (CDMA) Problems
13.5 A rate of 1/2 convolutional code with dmin = 10 is used to encode a data sequence occuring at
a rate of 1000 bits/s. The modulation is binary PSK. The DS spread spectrum sequence has a chip rate
of 10 MHz
a) Determine the coding gain
b) Determine the processing gain
c) Determine the Jamming margin assuming an Eb/J0 = 10
Solution:
a) The Coding gain is
Rc dmin = 1/2 ∗ 10 = 5 = 10log(5) = 6.989 db
b) The processing gain is W /R
W
R
=
107
2 ∗ 103
= 5 ∗ 10
3
= 10log(5 ∗ 10
3
) = 37 db
c) The Jamming margin is
Pav
Jav
=
(W /R)(Rc dmin)
(Eb/J0)
db
Pav
Jav
= (W /R)dB + (CGdB ) − (Eb/J0)dB
= 37 + 7 − 10 = 34dB

13.6 A total of 30 equal-power users are to share a common communication channel by CDMA. Each
user transmits information at a rate of 10 kbps via DS spread spectrum and binary PSK. Determine the
minimum chip rate in order to obtain a bit-error probability of 10−5
. Additive noise at the receiver may
be ignored in this computation.
Solution:
Pe = 10
−5
=
1
2
erfc


s
Eb
N0

 ∴ erfc


s
Eb
N0

 = 2 ∗ 10
−5
erf (u) = 1 − erfc(u) = 1 − 2 ∗ 10
−5
= 0.99998
From the error function table u =
p
Eb/N0 = 3.0 for a value of 0.99998
u =
s
Eb
N0
' 3.0 ∴ Eb/N0 = 9
To achieve an error probability of 10−5
, the required Eb/J0 = 10 Then, by using the relation in and we have
W /R
PN /PS
=
W /R
Nu − 1
=
Eb
J0
∴ W /R =
Eb
J0
(Nu − 1)
W = R
Eb
J0
(Nu − 1)
R = 104
bps, Nu = 30 and Eb/J0 = 10
W = 10
4
∗ 10 (30 − 1) = 2.9 ∗ 10
6
Hz
The minimum chip rate is 1/Tc = W = 2.9 ∗ 106
chips/sec

13.7 A CDMA system is designed based on DS spread spectrum with a processing gain of 1000 and binary
PSK modulation. Determine the number of users, if each user has equal power and the desired level of
performance is an error probability of 10−6
. Repeat the computation if the processing gain is changed to 500.
Solution:
Pe = 10
−6
=
1
2
erfc


s
Eb
N0

 i.e., 2 ∗ 10
−6
= erfc


s
Eb
N0


erfc(u) = 2 ∗ 10
−6
From the complementary error function table u =
p
Eb/N0 ' 3.36 for a value of 2 ∗ 10−6
∴ Eb/N0 = 11.3
Then, the number of users of the CDMA system is
Nu =
W /Rb
Eb/J0
+ 1
Nu =
W /Rb
Eb/J0
+ 1 = Nu =
1000
11.3
+ 1 = 89 users
If the processing gain is reduced to W /Rb = 500,then
Nu =
500
11.3
+ 1 = 45 users

13.8 A DS spread-spectrum system transmits at a rate of 1000 bps in the presence of tone interference. The
interference power is 20 dB greater than the desired signal and the required to achieve satisfactory
performance is 10 dB.
a Determine the spreading bandwidth required to meet the specifications.
b In the case of pulse interference, determine the pulse duty cycle that results in worst-case
performance and the corresponding probability of error.
Solution:
a (PJ /PS )dB = 20dB R = 1000 bps and (Eb/J0)dB = 10dB

W
R

db
=

PJ
PS

db
+

Eb
J0

db
= 30dB
W
R
= 1000
W = 1000R = 1000 ∗ 1000 = 10
6
Hz
b The duty cycle of pulse jammer for worst - case jamming is
α=
0.7
Eb/J0
=
0.7
10
= 0.07
The corresponding probability of error for this works case jamming is
P2=
0.082
Eb/J0
=
0.082
10
= 8.2 ∗ 10
−3

13.9 CDMA system consists of 15 equal power users that transmit information at a rate of 10,000 bits/s, each
using a DS spread spectrum signal operating at a chip rate of 1 MHz. The modulation is binary PSK.
a) Determine the Eb/Jo where Jo is the spectral density of the combined interference.
b) What is processing gain?
c) How much should the processing gain be increased to allow for doubling the number of users without
affecting the output SNR?
Solution:
a) We have Nu = 15 users transmitting at a rate of 10, 000 bps each, in a bandwidth of W = 1 MHz
The Eb/J0 is
Eb
J0
=
W /R
Nu − 1
=
106
/104
14
=
100
14
= 7.14(8.54dB)
b) Processing gain PG = Wc
Rb
1Mbps
10Kbps = 100
c) With Nu = 30 and Eb/J0 = 7.14, the processing gain should be increased to
W /R = (7.14)(29) = 207
Hence, the bandwidth must be increased to W = 2.07MHz

13.14 An m=10 ML shift register is used to generate the pseudorandom sequence in a DS SS. The chip
duration is Tc = 1microsec and the bit duration is Tb = NTc where N is the length of the m sequences
1 Determine the processing gain of the system in dB
2 Determine the jamming margin if the required E − b/J0 = 10 and the jammer is tone jammer with an
average power Jav
Solution:
1 The period of the maximum length shift register sequence is
N = 2
10
− 1 = 1023
Since Tb = NTc , then the processing gain is
N
Tb
Tc
= 1023(30dB)
2 According to jamming margin is
Jav
Pav dB
=
W
R dB
−
Eb
J0 dB
= 30-10 = 20dB
where Jav = J0W ≈ J0/Tc = J0 ∗ 106

13.15 An FH binary orthogonal FSK system employs a m = 15 stage linear feedback shift register that
generates a maximal length sequence. Each state of the shift register selects one of N nonoverlapping
frequency bands in the hopping pattern. The bit rate is 100 bits/sec and the hop rate is once/bit. The
demodulator employs noncoherent detection.
1 Determine the hopping bandwidth for this channel.
2 What is the processing gain?
3 What is the probability of error in the presence of AWGN?
Solution:
1 The length of the shift-register sequence is
L = 2
m
− 1 = 2
15
− 1 = 32767 bits
For binary FSK modulation, the minimum frequency separation is 2/T, where 1/T is the symbol (bit)
rate. The hop rate is 100 hops/ sec. Since the shift register has N = 32767 states and each state
utilizes a bandwidth of 2/T = 200 Hz, then the total bandwidth for the FH signal is 6.5534 MHz.
2 The processing gain is W/R. We have,
W
R
=
6.5534 ∗ 106
100
= 6.5534 × 10
4
bps
3 If the noise is AWG with power spectral density N0, the probability of error expression
P2 = Q


s
Eb
N0

 = Q
s
W /R
PN /PS
!

13.16 Consider the FH binary orthogonal FSK system described in Problem 13.15. Suppose that the hop rate
is increased to 2 hops/bit. The receiver uses squarelaw combining to combine the signal over the two hops.
1 Determine the hopping bandwidth for the channel.
2 What is the processing gain?
3 What is the error probability in the presence of AWGN?

Frequency Hopping Spread Spectrum Modulation Frequency Hopping Spread Spectrum Modulation
Frequency Hopping Spread Spectrum Modulation

The type of spread spectrum in which
carrier hops randomly from one frequency
to another is called Frequency Hop (FH)
spread spectrum.
Frequencies are shifted for every Tc
seconds.
Duration of signal element is Tb seconds.
FH/SS are of two types:
1 Slow Frequency Hopping (SFH):
2 Fast Frequency Hopping (FFH):
Data
modulator
d
BW W
=
0
2 cos( )
P w t
Binary
Data
Band pass
filter
0
2 cos( )
P w t
Phase
demodulator
Estimated
data
Frequency
synthesizer
Code
gnerator
FH code
clock
High
pass
filter
s
BW W
=
Frequency
synthesizer
Code
gnerator
FH code
clock
s
BW W
=
d
BW W
=
Figure: Transmitter
Data
modulator
d
BW W
=
0
2 cos( )
P w t
Binary
Data
Band pass
filter
0
2 cos( )
P w t
Phase
demodulator
Estimated
data
Frequency
synthesizer
Code
gnerator
FH code
clock
High
pass
filter
s
BW W
=
Frequency
synthesizer
Code
gnerator
FH code
clock
s
BW W
=
d
BW W
=
Figure: Receiver

Slow Frequency Hopping (SFH):
In slow frequency hopping the symbol rate of the input signal is an integer multiple of the frequency
hopping rate.
That is several symbols are transmitted on each frequency hop.
Slow FHSS has Tc ≥ Tb

Number of bits per MFSK symbol = 2 ⇒ M = 4
Rs = Rb/2 Rc = max(Rh, Rs) = Rs
Length of PN segment per hop = 3 Total number of frequency hops = 23
= 8

Rs = Rb/2 Rc = max(Rh, Rs) = Rs
= 8

Fast Frequency Hopping (FFH):
In fast frequency hopping the frequency hopping rate is an integer multiple of the input symbol rate.
That is the carrier frequency will change or hop several times during the transmission of the one symbol.
Fast FHSS has Tc Tb.

Rs = Rb/2 Rc = max(Rh, Rs) = Rh
= 8

Encoder
FSK
modulator
Mexer Channel Mexer
FSK
demodulator
Decoder
PN
sequence
generator
Frequency
synthesizer
Frequency
synthesizer
PN
sequence
generator
Time
sync
Information
sequence
Output
Figure: Block diagram of FH spread spectrum system
The initial modulation is usually binary or M-ary FSK.
FSK signal is translated by random frequency which is synthesized by frequency synthesizer from PN
codes.
Synthesized frequency is mixed with the output of the modulator.
The m bits from the PN generater will generate 2m
− 1 frequencies.
At the receiver an identical PN generator will generate m bits and these m bits will be used synthesize
corresponding 2m
− 1 frequencies.

Encoder Channel Decoder
PN
sequence
generator
Frequency
synthesizer
Frequency
synthesizer
PN
sequence
generator
Time
sync
Information
sequence
Output
Demodulator
k bits
m bits
Figure: Block diagram of an independent tone FH spread spectrum system
Independent tone hopping is less vulnerable to some jamming strategies.
In this m bits from the PN generator and the k bits are used to specify the frequency slot.
Synthesized frequency is mixed with the output of the modulator.
The FH rate is usually equal to or faster than symbol rate.
If there multiple hops it is called as fast hopped FHSS and if the hopping is performed at symbol rate is
called slow FHSS.

Time Hopping Spread Spectrum Modulation Time Hopping Spread Spectrum Modulation
Time Hopping Spread Spectrum Modulation

Encoder
Buffer
Innterleaver
Gate Gate
PSK
demodulator
PN
sequence
generator
PN
sequence
generator
Information
sequence
Output
Channel
PSK
modulator
Decoder
Time
sync
Buffer
Innterleaver
Figure: Block diagram of an independent tone FH spread spectrum system
In time hop (TH) a time interval which is selected to be much larger than the reciprocal of the
information rate.
The coded information are transmitted in pseudorandomly selected time slot as a block of one or more
code words.
The PSK modulation is used to transmit the coded bits.
Because of the burst characteristics buffer storage is used at the transmitter.
Buffer storage is used at the receiver to provide a uniform data stream to the user.

In an FH system an FH tone of short duration is referred as a chip.
The chip rate Rc for an FH system is defined by:
Rc = max(Rh, Rs )
where Rh is the hop rate and Rs is symbol rate
Let J be the average Jammers signal over the entire frequency hopped spectrum of bandwidth Wc Hz,
then the average power spectral density N0/2 of the equivalent AWGN is:
N0
2
=
1
2
J
Wc
N0 =
J
Wc
Thus, the average energy E to noise density ratio is
E
N0
=
P/J
Wc /RS
Where P/J is reciprocal of the jamming margin

Synchronization Synchronization
Synchronization

Synchronization Synchronization
Synchronization
In spread spectrum system, de-spreading is done in order to retrieve the
message signal. [3, 4, 5, 6].
In order to de-spread the received signal, the locally generated receiver
spreading code must be in synchronous to the transmitter spreading code.
Time synchronization of the receiver to the received spread-spectrum signal
may be achieved in two distinct phases:
1 Initial Acquisition (Coarse synchronization).
2 Tracking (Fine synchronization).

Synchronization Phase for Direct sequence spread spectrum:
Phase for Direct sequence spread spectrum:
The initial synchronization is the one in which synchronize the receiver clock to the transmitter clock.
There is always an initial timing uncertainty that is due to propagation delay in the transmission of the
signal through the channel.
The initial synchronization is achieved by transmitting a known pseudorandom sequence to the receiver.
The receiver is continuously in a search mode looking for this sequence in order to establish initial
synchronization.
Suppose that the initial timing uncertainty is Tu seconds and the chip duration is Tc .
Dwell time Td = NTc is required to test synchronism at each time instant.
If the search over the time uncertainty interval in (coarse) time steps of Tc /2, then the time required to
establish initial synchronization is
Tinit sync =
Tu
Tc /2
NTc = 2NTu
The transmitted sequence must be at least as long as 2NTu in order to perform necessary search .
Rc
(t)
0 0 1 1 1 0 1 0 0 1 1 1 0 1
Tc NTc
Tc
Tc
Figure: Plot of correlation.

Synchronization Carrier Synchronization
Serial Search sliding correlator:
The correlator cycles in Tc /2 discrete-time intervals.
The crosscorrelation is performed over NTc interval
where N is the number of chips, and the correlator output is
compared with a threshold to determine if the known signal
sequence is present.
If the threshold is not exceeded, the known reference se-
quence is advanced by Tc /2 sec and the correlation process
is repeated.
These operations are performed until a signal is detected.
Consider a BPSK modulated DS-SS signal
v(t) = g(t)s(t) =
p
2Ps g(t)d(t) cos w0t
where g(t) is spreading code, d(t) binray baseband data and
w0 is IF carrier frequency During initial synchronization, the
baseband data signal is set to constant value d(t)=1.
v(t) =
p
2Ps g(t) cos(w0t + θ)
Prior to acquisition, the transmit and receive PN codes are
not in synchronism, that is, the time position of the chip
patterns is not aligned. The output of the multiplier is given
by
vr (t) = vi (t).g(t − iTc )
=
p
2Ps g(t) cos(w0t + θ).g(t − iTc )
PN Code
generator
Threshold
Detector
Search
control
clock
Received
Signal
( )
0
Tc
dt
λ
∫
Integrator
( )
c
g t iT
−
0
2 ( ) ( )cos( )
s
P g t d t w t θ
+
( )
r
v t
Figure: A sliding correlator.
The repetitive trial-and-error process eventually
leads to a state in which the received chipped
signal g(t) is aligned with g(t − iTc ) that is, the
relative shift.
g(t).g(t − iTc ) = g(t).g(t − 0.Tc ) = 1
This is the received ”pilot” signal is despread
and becomes
vr (t) =
p
2Ps cos(w0t + θ)
The integrator output transfers the complete re-
ceived signal power to the envelope detector.
This leads to a high comparator output state,
and acquisition or crude synchronization is now
established.

Synchronization Symbol Synchronization
Tracking:
The tracking maintains the PN code generator at the receiver in synchronism with the received signal
i.e., fine-chip synchronization.
In this tracking loop, the received signal is applied to two multipliers, where it is multiplied by two
outputs from the local PN code generator which are delayed relative to each other by an amount of
2δ ≤ Tc
The product signals are the crosscorrelations between the received signal and the PN sequence at the
two values of delay.
vD (t) =
p
2Ps d(t)g(t)g(t+τ−Tc /2) cos(w0t+θ) vA(t) =
p
2Ps d(t)g(t)g(t+τ+Tc /2) cos(w0t+θ)
vDF (t) =
p
2Ps d(t) cos(w0t+θ)[g(t)g(t + τ − Tc /2)] vAF (t) =
p
2Ps d(t) cos(w0t+θ)[g(t)g(t + τ + Tc /2
Bandpass
Filter
Local
oscillator
VCO
Envelope
detector
Loop
Filter
Received
Signal
PN code
generator
Advance by
δ
Retard by
δ
Bandpass
Filter
Envelope
detector
To demodulator
+
−
( )
y t
( / 2)
c
g t T
τ
+ −
( / 2)
c
g t T
τ
+ +
A
V
D
V DF
V
AF
V
0
( ) 2 ( ) ( )cos( )
i s
v t P g t d t w t θ
= +
Figure: Simple PLL.

The average value of the product of the received PN sequence and a shifted version of the same
sequence is the autocorrelation function defined by
Rg (τ ± Tc /2) = g(t).g(t + τ ± Tc /2)
The envelope detectors extract the envelopes of and and removes the data . Then
VD (t) = |Rg (τ − Tc /2)| VA(t) = |Rg (τ + Tc /2)|
The input to VCO is y(t) given by
y(t) = |Rg (τ − Tc /2) − Rg (τ + Tc /2)|
If τ is positive, a positive voltage appears at the VCO input and the VCO frequency is increased.
This increased VCO frequency reduces τ.
For negative values of τ a negative y(t) voltage is generated at the VCO input decreasing the VCO rate
and thereby increasing τ .
0
| ( / 2) |
g c
R T
τ −
| ( / 2) |
g c
R T
τ
− +
c
T
2
c
T
2
− c
T
c
T
c
3T
2
c
3T
2
− τ
( ) | ( / 2) | | ( / 2) |
g c g c
y t R T R T
τ τ
= − − +
c
3T
2
c
T
2
c
T
2
−
c
3T
2
− τ
Figure: Autocorrelation function and tracking error signal for DLL.

Tau-dither loop (TDL):
Tau-dither loop (TDL), employs only a single “arm” instead of the two “arms”.
In this case, the crosscorrelator output is regularly sampled at two values of delay, by stepping the code
clock forward and backward in time by an amount δ.
The envelope of the crosscorrelation that is sampled at ±δ has an amplitude modulation whose phase
relative to the tau-dither modulator determines the sign of the tracking error.
The error signal is low pass filtered and then applied to a voltage controlled oscillator that controls
(symbol waveform generator) the charging and discharging instants of the correlators.
The instantaneous frequency of the local clock is advanced or retarded in an iterative manner until the
equilibrium point is reached, and symbol synchronization is thereby established.
Bandpass
Filter
Local
oscillator
VCO
Envelope
detector
Loop
Filter
Received
Signal
PN code
generator
Tau-dither
generator
Figure: Tau-dither loop (TDL).

Synchronization Acquisition Phase for FH spread-spectrum system
Acquisition Phase for FH spread-spectrum system:
Synchronize the PN code sequence generated at the receiver that controls the hopped frequency pattern.
Bank of matched filters tuned to the transmitted frequencies in the known pattern may be employed.
Their outputs properly delayed, envelope or square-law detected, and added to produce the signal
output which is compared with a threshold.
A signal present (signal acquisition) is declared when the threshold is exceeded.
The search process is usually performed continuously in time until a threshold is exceeded.
Filter
Tuned to f1
Filter
tuned to f3
Filter
tuned to f2
Envelope
detector
Delay one
(chip)
Envelope
detector
Envelope
detector
Delay
Filter
tuned to fM
Envelope
detector
Delay
Threshold
detector
Sync
pulse
Received
Signal
Figure: Tau-dither loop (TDL).

Synchronization Acquisition Phase for FH spread-spectrum system
Acquisition Phase for FH spread-spectrum system:
It is based on a serial search and is similar to the sliding correlator for DS spread-spectrum signals.
It is of single matched-filter and an envelope detector preceded by a frequency hopping pattern
generator and followed by a threshold detector.
Tuned
Filter
Envelope
detector
Postdetection
integration
Threshold
detector
Sync
pulse
Received
Signal
PN code
sequence
generator
Frequency
synthesizer
Clock
Figure: Alternative for acquisition for FH signal.

References
http://www.women-inventors.com/Hedy Lammar.asp.
http://people.seas.harvard.edu/ jones/cscie129/nu lectures/lecture7/hedy/lemarr.htm.
S. Haykin, Digital communications, 2nd ed. Wiley, 1988.
J. G. Proakis, Digital communications, 4th ed. Prentice Hall, 2001.
J. G. Proakis and M. Salehi, Communication Systems Engineering, 2nd ed. Prentice Hall, 2002.
Forouzan and A. Behrouz, Data Communications and Networking, 4th ed. Tata-McGraw-Hill, 2007.

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